The problem below was presented earlier today and I am wondering why a "stars and bars" approach wouldn't be an appropriate method to solving this problem. The problem reads as follows:
Find the probability that each child gets at least 1 ball when we are distributing 5 DISTINCT balls among 4 children (who are distinct of course).
Here is how I interpreted this problem.
Counting the number of ways we can distribute $n=5$ balls among $k=4$ children in such a way so that each child gets at least one ball is equivalent to counting the number of ways we can express $n=5$ as a sum of $k=4$ positive integers, of which there are nCr($5-1$,$4-1$) ways via stars and bars.
Meanwhile, the number of ways we can distribute the $n=5$ balls among the $k=4$ children in any way we'd like is equivalent to counting the number of ways we can express $n=5$ as a sum of $k=4$ non$-$negative integers, of which there are nCr($5+4-1$,$5$) ways.
Dividing these two numbers gives us a probability of $\frac{1}{14}$ which is incorrect.
Is my problem with this approach that the "star and bars" approach doesn't account for the fact that the balls are distinct?
Yes, that’s the problem. If the children are $A$, $B$, $C$, and $D$, there are actually $\binom52\cdot 3!=60$ different ways to distribute the balls so that $A$ gets $2$ balls and $B$, $C$, and $D$ get one each: there are $\binom52=10$ different pairs of balls that can be given to $A$, and the remaining $3$ balls can be permuted amongst $B,C$, and $D$ in $3!=6$ distinguishable orders.